Low Cost, High Performance Nutation Damper for a Spinning Spacecraft
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Low Cost, High Performance Nutation Damper For a Spin-Stabilized Spacecraft
Mark A. Woodard NASA/GSFC
Introduction
Spacecraft designers often employ spin-stabilization as the means of spacecraft attitude (pointing) control. The approach is to deploy the spacecraft from the launch vehicle with angular momentum near or along the spacecraft major moment of inertia. This provides a stabilization mode that is very insensitive to small perturbing torques. However, larger torques – such as those imparted from the launch vehicle at separation or from thrusters during precession and orbit maneuvers – can impart transverse momentum on the spacecraft that will cause the spacecraft to wobble, or nutate. It is usually desirable to reduce any spacecraft nutation that arises from such perturbations in order to return the spacecraft to a pure spin-stabilized mode.
Nutation is often reduced by attaching a passive nutation damper to the spacecraft. The passive nutation damper is a self-contained motion device that will attenuate transverse momentum by dissipating excess energy through friction, such as a fluid moving through a closed ring. Two approaches are commonly used. The first approach is to use a closed ring that is partially filled with a fluid. By only partially filling the ring, the fluid has room to expand if temperature increases on-orbit – however, this design has some disadvantages - the fluid slug may lock up inside the ring at low spin rates or low nutation angles, thus rendering the damper ineffective; also, this type of damper must be tuned to a certain spin rate and are not as effective as the spin rate varies. The second commonly used approach is to fully fill the ring with fluid and attach a bellows device to the ring that gives the fluid room to expand and contract, so that an increase in temperature does not create high pressure inside the ring which could lead to bursting. The disadvantage of this approach is that a bellows can be a long lead and expensive component, adds additional mass, and requires additional fabrication and testing.
A novel approach is to build a nearly filled ring with an escape tube instead of a bellows. The escape tube allows the ring to remain fully filled with fluid even as the spacecraft and damper go through a wide range of dynamic variations and temperature variations. By careful design and placement of the escape tube, any voids that may form in the fluid will gravitate towards the end of the escape tube, thus keeping the ring fully filled with fluid while allowing room for fluid expansion and contraction with only slight changes in internal pressure. This approach will provide a low cost damper that provides high reliability and high performance across a broad range of operating conditions. Basic Nutation Damper Design
The basic nutation damper design is shown in Figure 1. The damper consists of 5 components: 1. A Ring, formed from rigid but bendable tubing or pipe 2. An Escape Tube, also formed from rigid tubing or pipe 3. A 3-way Tee Fitting – the two opposing ports accept either end of the ring, while the third port accepts one end of the escape tube 4. A Cap, used to seal the damper 5. Damper Fluid, used to fill the entire ring, Tee fitting, and roughly half of the escape tube.
Figure 1. Nutation Damper Design
The damper coordinates are defined as follows: The Y axis is normal to the plane of the ring The Z axis is in the plane of the ring and defines the “UP” direction of the damper The X axis completes the right-hand triad
The radius of the ring is labeled as R, and is measured from the center of the ring to the centerline of the tubing. The escape tube bend angle is labeled as , and is measured as the angle between the Z axis and the escape tube axis. The damper should be mounted along one of the side walls of the spacecraft, such that the damper Y axis is nearly perpendicular to the nominal spacecraft spin (momentum) axis. By properly orienting the damper Z axis, we can assure that any gaseous voids that form inside the damper ring will gravitate towards the cap end of the escape tube, thus leaving the ring fully filled with fluid. The damper Z-axis should be oriented such that it is aligned as closely as possible with the zenith axis while the launch vehicle is on the launch pad. This will ensure that any bubbles will rise to the top of the escape tube before launch, since gravitational forces are in the –Z direction. The damper Y-axis should be oriented towards the center of the spacecraft. In the microgravity of space, the dominant force acting on the fluid will be the centrifugal force, which will be in the –Y direction, thus causing any gaseous voids to move to the end of the escape tube. Once on orbit, the damper ring will remain fully filled with fluid as long as the spacecraft is spinning, which is our design goal.
Another important consideration in damper mounting is to mount the damper as close as possible to the spacecraft intermediate principal axis of inertia (I2), as this location will provide the maximum dynamic motion during the nutation cycle. Figure 2 illustrates the orientation of the damper axes (X, Y, Z) relative to the spacecraft principal axes (I1, I2, I3). Note that the damper is tilted inward (in a right-hand sense about the –X axis) by an angle , such that the tee fitting is the portion of the damper ring closest to the spacecraft center of mass. With the damper tilted inward, centrifugal acceleration will induce any gaseous voids in the damper fluid to migrate towards the tee fitting and then the escape tube. The parameter L is the distance from the damper plane to the spacecraft center of mass (c.m.) and is the spacecraft spin rate about the I3 axis.
Figure 2. Nutation Damper Orientation Relative to Principal Axes
Nutation Damper Performance
Define: R = radius of the damper ring r = tubing inside radius = fluid viscosity = fluid density = fluid surface tension M = fluid mass in damper ring = spin rate I1, I2, I3 = spacecraft principal moments of inertia (1 = minor axis, 2 = intermediate axis, 3 = major axis) = damper tilt angle L = distance from spacecraft center-of-mass to damper center-of-mass
The equations that model the nutation damper performance are as follows.
The damper circumference at the center of the tubing cross-section is
C = 2R (1)
The fluid mass is
M = r2C = 22Rr2 (2)
The nutation frequency is
1/2 = [(I3-I1)(I3-I2)/(I1I2)] (3)
The Wobble Reynolds Number (WRN), , is a ratio of the fluid momentum and fluid viscosity, and is computed as
= r2/ (4)
The value of the WRN is critical for optimizing the nutation energy dissipation per unit mass of fluid. Figure 3 illustrates the importance of properly tuning the WRN. The nutation damper dissipates energy most efficiently when the WRN is in the range of 4 to 10, or when the fluid momentum force is slightly higher than the viscous force. This maintains the normalized energy dissipation per cycle, , in the range of 1.0 to 1.2. Figure 3. Energy Dissipation vs. Wobble Reynolds Number.
The nutation damper time constant is a measure of the time required to damp the nutation angle by a factor of 1/e. The characteristic equation governing nutation angle damping as a function of time from an initial condition is
(-t/) (t) = (t0)e (5) where t is time, is the nutation angle, and is the nutation damping time constant. The time constant is computed as
2 2 = 2I2 /(MR I3cos) (6a) or from eq. 2,
2 3 2 = I2 /(R r I3cos) (6b)
In addition, the damper ring must be properly sized such that the centrifugal force arising from the spacecraft spin will dominate the surface tension force inherent in the damper fluid. This will ensure that any voids (bubbles) than are contained in the fluid will migrate into the end of the escape tube under the influence of centrifugal force. An important dimensionless number used in fluid dynamics is the “Bond Number”, which is defined as the ratio of acceleration force to surface tension force, and is desired to be >>1. For a circular nutation damper, the Bond Number is defined as
2 2 NB = r L / (7)
Once the spacecraft parameters (L and ) and fluid properties ( and ) are known, we can increase the Bond number to be >>1 simply by increasing the tubing inside radius, r, to avoid fluid lockup. Then, using eq. 6b, we can solve for the required damper ring radius, R, to achieve a desired damping time constant, , for a given spacecraft inertia set (I2, I3). Once a suitable time constant is found, the WRN can be tuned by varying the fluid viscosity. Note that there is a trade-off between damper performance reliability and damper mass – increasing r increases the Bond number but also increases the fluid mass proportionally, since both terms go as r2. Temperature Effects
A survey of various fluids indicates that dimethyl silicone fluid is an excellent choice for the nutation damper for several reasons: It is commercially available in a wide range of viscosities Its viscosity changes relatively little with temperature, compared to other fluids It has a low coefficient of thermal expansion (~0.001/C) It is essentially nontoxic and requires no special handling It is extremely inert and non-reactive with most other materials It is stable at high temperatures
The physical properties of dimethyl silicone at 25C are given in Table 1.
Viscosity Density Surface Coefficient of Freezing Point (cs) (gm/cm3) Tension (N/m) Expansion (1/C) (C) 5.0 0.913 0.0197 0.00105 -70 10.0 0.935 0.0201 0.00108 -60 20.0 0.949 0.0206 0.00107 -52 Table 1. Physical Properties of Dimethyl Silicone Fluid.
Note that 1 centistoke = 10-6 m2/sec. Also note that the term “viscosity” as used in this analysis refers to the “kinematic” viscosity, not the “absolute” viscosity. Kinematic viscosity is defined as the ratio of the fluid absolute viscosity (measured in units of centiPoise) and the fluid density, or k = a/. The absolute viscosity can be determined simply by multiplying the kinematic viscosity by the specific gravity
Temperature affects fluid expansion and contraction, and should be considered when designing the length of the escape tube. For dimethyl silicone fluids, the coefficient of thermal expansion is approximately 0.001/C, as shown in Table 1. So for example, if the expected operational/survival temperature range is expected to vary +/- 50C from ambient, one would expect a +/-5% expansion or contraction of the fluid in the ring. Assuming that the same diameter tubing is used in both the ring and escape tube and that the escape tube is half filled with fluid at ambient temperature, the escape tube must be at least 10% as long as the ring circumference, or 20% as long to provide for a margin factor of 2.
The operational temperature limits of the nutation damper must be analyzed to ensure that the damper will perform effectively across the entire range of temperatures. Variations in temperature will also affect viscosity, WRN, energy dissipation, and ultimately the damper time constant, as described in eqs. 4 and 6. The viscosity reference temperature of the Dow Corning 200 silicone fluid is +25C. The relationship between viscosity and temperature is given in the equation below and illustrated in Figure 4:
log10[log10(+0.7)] = A - Blog10(T+273.15) (8) where
= kinematic viscosity [cs] T = temperature [C] A and B are dimensionless constants:
A 7.172, B 2.948 for 5 cs fluid A 5.671, B 2.287 for 10 cs fluid A 5.021, B 1.981 for 20 cs fluid Figure 4. Viscosity vs. Temperature for Silicone Fluid
It is important to note that with a properly sized escape tube, variations in temperature will have a negligible effect on internal pressure of the nutation damper, since any gasses trapped in the escape tube will act as a compressible fluid. This is a significant improvement over previously designed fully-filled dampers, where the damper fluid will behave as a compressible fluid and large internal pressures can arise from small increases in temperatures. The proposed damper design is therefore safer and more reliable than a fully filled damper without an escape tube; it is also a simpler and more robust design that a damper with an attached bellows.
The bending angle, , of the escape tube must be between 0 and 90 in order to ensure that the escape tube has components in both the +Y and +Z directions, as mentioned earlier. It is important that (+) be greater than the maximum expected spacecraft nutation angle so that the escape tube cap is always the innermost part of the damper assembly – this will ensure that centrifugal forces on the damper fluid are benign for large nutation angles. Prelaunch Testing
A nutation damper test apparatus exists at NASA GSFC. It consists of a platform and test equipment suspended from a torsional pendulum. The platform is deflected and allowed to rotate back and forth. By attaching the nutation damper to the platform and measuring how rapidly the rotation is attenuated, we can estimate the nutation damper time constant. An important consideration is that the damper Y-axis must be pointed up during testing such that gravitational force is in the same direction as centrifugal force will be once in orbit. In the ground testing environment, the Bond number equation is expressed as
2 NB = r g/ (9)
Where g is the gravitational acceleration, 9.81 m/sec2. References 1. Dow Corning 200 Product Information, Dow Corning Corporation, 1997. 2. FAST Nutation Damper Spread Sheet Description, D. Mangus, October 1991. 3. Final Report for Study on Fluid Dampers for Booms, L.R. Koval et al, July 1968. 4. Silicones and Their Uses, R. McGregor, 1954. 5. ST5 Nutation Damper Performance Testing Procedure, J. Morrissey, February 2003. 6. Surface Tension Lockup in the IMAGE and Lunar Prospector Nutation Dampers, C. Hubert, June 2000. 7. The Handbook of Fluid Dynamics, R. W. Johnson, 1998. 8. Tribology Data Handbook, E. R. Booser, 1997.